What is the integral of x^2*sin(ln(x)) ?

The integral of x^2*sin(ln(x)) is -1/10 x^3cos(ln(x))+3/10 x^3sin(ln(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^2*sin(ln(x))

\int x^{2} \cdot sin (ln (x)) d x

- \frac{1}{10} x^{3} cos (ln (x)) + \frac{3}{10} x^{3} sin (ln (x)) + C

Solution steps

\int x^{2} sin (ln (x)) d x

Apply u-substitution

: \int e^{3 u} sin (u) d u

= \int e^{3 u} sin (u) d u

Apply Integration By Parts

: - e^{3 u} cos (u) - \int - 3 e^{3 u} cos (u) d u

= - e^{3 u} cos (u) - \int - 3 e^{3 u} cos (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{3 u} cos (u) - (- 3 \cdot \int e^{3 u} cos (u) d u)

Apply Integration By Parts

: e^{3 u} sin (u) - \int e^{3 u} \cdot 3 sin (u) d u

= - e^{3 u} cos (u) - (- 3 (e^{3 u} sin (u) - \int e^{3 u} \cdot 3 sin (u) d u))

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - e^{3 u} cos (u) - (- 3 (e^{3 u} sin (u) - 3 \cdot \int e^{3 u} sin (u) d u))

Therefore

\int e^{3 u} sin (u) d u = - e^{3 u} cos (u) - (- 3 (e^{3 u} sin (u) - 3 \cdot \int e^{3 u} sin (u) d u))

Isolate

\int e^{3 u} sin (u) d u

= - \frac{e ^{3 u} cos ( u )}{10} + \frac{3 e ^{3 u} sin ( u )}{10}

Substitute back

u = ln (x)

= - \frac{e ^{3 l n (x)} cos ( ln ( x ) )}{10} + \frac{3 e ^{3 l n (x)} sin ( ln ( x ) )}{10}

Simplify

- \frac{e ^{3 l n (x)} cos ( ln ( x ) )}{10} + \frac{3 e ^{3 l n (x)} sin ( ln ( x ) )}{10} : - \frac{1}{10} x^{3} cos (ln (x)) + \frac{3}{10} x^{3} sin (ln (x))

= - \frac{1}{10} x^{3} cos (ln (x)) + \frac{3}{10} x^{3} sin (ln (x))

Add a constant to the solution

= - \frac{1}{10} x^{3} cos (ln (x)) + \frac{3}{10} x^{3} sin (ln (x)) + C

\int \frac{x ^{4}}{3} + \frac{5}{x ^{3}} d x

\int (\frac{( y ^{4} - 1 )}{y})^{2} d y

\int \frac{2}{( x + 1 ) ( 3 x + 5 )} d x

\int 6 - 4 x d x

\int 12 x^{\frac{5}{7}} + 5 x^{- \frac{6}{7}} d x

What is the integral of x^2*sin(ln(x)) ?
The integral of x^2*sin(ln(x)) is -1/10 x^3cos(ln(x))+3/10 x^3sin(ln(x))+C